Here are three different platforms one can have when planning to put in place a coaching or training business that needs to deliver online training.
Everything on your brand/website.
This is an obvious solution. Every business wants to keep their brand front and center!
So, delivering your course content from your website is ideal but requires more effort.
You or your dev person/agency has to ensure everything is up and running 24/7.
If your main website is on WordPress CMS, then implementing a learning management system in WP is possible with the many plugins readily available.
A lot of these LMS plugins are quite robust and feature rich so getting up and running can be relatively quick.
Host your course on 3rd party platform that provides some branding flexibility.
Platforms like Teachable, Kajabi and ClickFunnels provide what we call 'software as a service' or SaaS solutions.
You can have a sub-domain pointed to their platform and your clients and students login and get access to your course.
You can reap immediate benefits and can even get started for free, but loose some ability to customize things as you would be limited by the features and functionality provided by these platforms.
The trade off between SaaS option and Dev team/agency maintaining your website is that SaaS is usually a bit more cost effective. But it's possible to find some good dev team/agency for about the same investment.
Third party course platforms or marketplaces.
One can even become an instructor on a platform like Udemy where they host and promote your course.
Here you would be giving up a lot but in return you get free traffic and students.
What you give up is some pricing control and even student contact information is not shared. So in my opinion you are not really building your business, you are building their business in the long term.
Still, this is a viable option if you have a strategy and you have 'creative' systems put in place to get your students on "your platform" or "email list" somehow.
So, as you can see, there are lots of options when it comes to how you implement your tech stack to build your coachig/training business.
Based on your strategy and plan you can pick one or all to grow your business.
Here is the link to the 6 part video series on domain name and hosting